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  • Find the value of limit: <img alt="\lim_{x\rightarrow \infty }\left ( \sqrt{100x^{2}+7x}-10x \right )" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim_%7Bx%5Crightarrow%20%5Cinfty%20%7D

Find the value of limit: \lim_{x\rightarrow \infty }\left ( \sqrt{100x^{2}+7x}-10x \right )

Option: 1

\frac{7}{2\sqrt{10}}


Option: 2

\frac{7}{\sqrt{10}}


Option: 3

\frac{7}{2}


Option: 4

Limit does not exists.


Answers (1)

For solving the limits of the type ∞ - ∞, we simply rationalize the given limit, that is multiplying and dividing the limit by its additive inverse.

Additive inverse of \lim_{x\rightarrow \infty }\left ( \sqrt{100x^{2}+7x}-10x \right ) is \lim_{x\rightarrow \infty }\left ( \sqrt{100x^{2}+7x}+10x \right )

Hence,

\begin{aligned} & \lim _{x \rightarrow \infty} \frac{\left(\sqrt{100 x^2+7 x}-10 x\right)\left(\sqrt{100 x^2+7 x}+10 x\right)}{\left(\sqrt{100 x^2+7 x}+10 x\right)} \\ &= \lim _{x \rightarrow \infty} \frac{100 x^2+7 x-100 x^2}{\left(\sqrt{100 x^2+7 x}+10 x\right)} \\ &= \lim _{x \rightarrow \infty} \frac{7}{\left(\sqrt{100+\frac{7}{x}}+10\right)} \\ &= \frac{7}{2 \sqrt{10}} \end{aligned}

 

 

Posted by

Kshitij

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